3/6 AN INTRODUCTION TO CYBERNETICS 



Ex. 4: (Continued.) How would the graph alter if further operands were 

 added? 



Ex. 5: Find the transform of (3,-2,1) by A if the general form is (^,A,y) and 

 the transformation is 



[g' = 2g-h 

 A:^h'= h-j 



ir = g + h 



Ex. 6: Arthur and Bill agree to have a gamble. Each is to divide his money 

 into two equal parts, and at the umpire's signal each is to pass one part 

 over to the other player. Each is then again to divide his new wealth 

 into two equal parts and at a signal to pass a half to the other; and so on. 

 Arthur started with 8/- and Bill with 4/-. Represent the initial operand 

 by the vector (8,4). Find, in any way you can, all its subsequent transforms. 



Ex. 7: (Continued.) Express the transformation by equations as in Ex. 5 

 above. 



Ex. 8: (Continued.) Charles and David decide to play a similar game except 

 that each will hand over a sum equal to a half of what the other possesses. 

 If they start with 30/- and 34/- respectively, what will happen to these 

 quantities? 



Ex. 9: (Continued.) Express the transformation by equations as in Ex. 5. 



Ex. 10: If, in Ex. 8, other sums of money had been started with, who in general 

 would be the winner? 



Ex. 11 : In an aquarium two species of animalcule are prey and predator. In 

 each day, each predator destroys one prey, and also divides to become two 

 predators. If today the aquarium has m prey and // predators, express 

 their changes as a transformation. 



Ex. 12: (Continued.) What is the operand of this transformation? 



Ex. 13: (Continued.) If the state was initially (150,10), find how it changed 

 over the first four days. 



Ex. 14: A certain pendulum swings approximately in accordance with the 

 transformation x' — \(x — y), y' — \{x + ;'), where x is its angular 

 deviation from the vertical and y is its angular velocity; x' and y' are 

 their values one second later. It starts from the state (10,10); find how its 

 angular deviation changes from second to second over the first eight 

 seconds. (Hint: find x', x", x'", etc.; can they be found without cal- 

 culating y', y", etc?) 



Ex. 1 5 : (Continued.) Draw an ordinary graph (with axes for x and /) showing 

 how .v's value changed with time. Is the pendulum frictionless ? 



Ex. 16: In a certain economic system a new law enacts that at each yearly 

 readjustment the wages shall be raised by as many shillings as the price 

 index exceeds 100 in points. The economic effect of wages on the price 

 index is such that at the end of any year the price index has become equal 

 to the wage rate at the beginning of the year. Express the changes of 

 wage-level and price-index over the year as a transformation. 



Ex. 17: (Continued.) If this year starts with the wages at 110 and the price 

 index at 1 10, find what their values will be over the next ten years. 



Ex. 18: (Continued.) Draw an ordinary graph to show how prices and wages 

 will change. Is the law satisfactory? 



34 



